Properties

Degree $2$
Conductor $24$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.577·3-s + 1.54·5-s − 0.318·7-s + 0.333·9-s + 0.550·11-s − 1.02·13-s − 0.892·15-s − 0.432·17-s + 1.08·19-s + 0.183·21-s − 0.205·23-s + 1.39·25-s − 0.192·27-s + 0.809·29-s − 0.819·31-s − 0.317·33-s − 0.492·35-s − 1.74·37-s + 0.591·39-s + 0.159·41-s + 0.444·43-s + 0.515·45-s − 0.354·47-s − 0.898·49-s + 0.249·51-s + 1.16·53-s + 0.851·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s+1.91i) \, \Gamma_{\R}(s-1.91i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 24,\ (1.91464670089i, -1.91464670089i:\ ),\ 1)\)

Euler product

\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line